The England international broke the deadlock shortly after half-time at Stamford Bridge, driving through the visitors' defence before lifting a composed finish over goalkeeper Senne Lammens. Cole Palmer subsequently converted a penalty to secure three points for Chelsea.
Despite already contributing six goals and four assists in all competitions this season, Rogers believes his finishing remains an area where he can make significant progress.
"Being more clinical in front of goal, being more ruthless in and around the box, is something that previously I was a bit disappointed in myself because I'd score a nice goal every once in a while, but I wouldn't score the easy goals," he said.
"I'm trying to work on that here. I'm trying to improve in my final actions, so that's my biggest one. I'm an attacker, so I try to score goals and assist and create chances, and I want to keep improving. I'm not there yet, but it's a good start!"
His goal against United demonstrated the individual quality he can bring to Chelsea's attack. Receiving a pass from Romeo Lavia in midfield, Rogers drove towards the penalty area, initially considering a pass to the supporting Estevao before recognising the space opening in front of him.
"I knew it was a counter-attacking opportunity to drive, and I think Estevao made the run on the outside. I was just going to keep driving and I was looking to play him, but then the space just kind of opened up and I just kept going," he explained.
"Then it was just about being composed in front of goal and luckily it went in."
Rogers was pleased to find the net, particularly given the timing of the breakthrough after a difficult first half for the hosts. But he placed greater importance on the contribution his finish made to the team's result.
"It was nice to see it go in the back of the net at an important time in the game," he added.
"So that's the most important thing for me, that I can make a difference to the game and ultimately we go on and win the game because of it, so I'm happy."
Having joined Chelsea after two years at Aston Villa, Rogers is enjoying the challenge of establishing himself in a new squad. He believes the team's progress will depend on building collective understanding and maintaining standards through the difficult periods of a season.
"I'm loving it. It's a different challenge, a new challenge, and I'm really enjoying playing with these wonderful quality players.
"It's just about the team coming together as a whole. There are some good times, bad times, but I've just got to be a bit of a leader, stick to what I know and do the best I can. So we just need to keep going."